Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-14T07:55:39.210Z Has data issue: false hasContentIssue false
  • English
  • Français

White noise approach to Gaussian random fields

Published online by Cambridge University Press:  22 January 2016

Ke-Seung Lee
Ke-Seung Lee*
Affiliation:
College of Liberal Arts and Sciences Department of Mathematics Korea University, Chochiwan, Korea
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The purpose of this paper is to investigate way of dependency of Gaussian random fields X(D) indexed by a domain D in d-dimensional Euclidean space Rd . Our main tool is variational calculus, where the boundary of a domain varies and deforms and we appeal to the white noise analysis. We therefore assume that X(D) is expressed white noise integral of the form

(0.1) X(D) = X(D, W)=∫D F(D, u)W(u)du,

where W is the Rd -parameter white noise and the kernel F(D, u) is a square integrable function over Rd , and where D is a bounded domain with smooth boundary.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 119 , September 1990, pp. 93 - 106
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1990

References

[1] Levy, P., Processus stochastiques et mouvment brownein, Gauthier-Villars, Paris, 1948.Google Scholar
[2] Levy, P., A special problem of Brownian motion and a general theory of Gaussian random function, Proc. 3rd Berkeley Symp., Vol. II (1956), 133–175.Google Scholar
[3] Gel’fand, I. M. and Naimark, M. A., Unitary representation of the group of linear transformations of the straight line, Comptes Rendus (Doklady) de l’Acade’mie des Sciences de l’URSS (1947), Vol. 55, No. 7, 567–570.Google Scholar
[4] Hida, T., Canonical representation of Gaussian process and their applications, Memo. Coll. Sci., Univ. of Kyoto, 33 (1960), 109–351.Google Scholar
[5] Hida, T. Brownian Motion (in Japanese), Iwanami Pub. Co., Tokyo (1975); English Trans. Springer—Verlag, New York, Heidelberg, Berlin (1980); Russian Trans., Nauka, Moscow, 1987.Google Scholar
[6] Hida, T., Analysis of Brownian functionals, Carleton Mathematical Lecture Notes, no. 13 (1975), and 2nd ed. (1978).Google Scholar
[7] Hida, T., A note on generalized Gaussian random fields, Jour. Multivariate Analysis, 27 (1988), 255–260.Google Scholar
[8] Hida, T., White noise analysis and Gaussian random field, Proc. 24th Karpacz Winter School of Theoretical Physics, 1988, Stochastic Methods in Mathematics and Physics. World Scientific, 277–289.Google Scholar
[9] Hida, T., White noise and Gaussian random fields, Proc. Singapore Probability Conference, June 1989, ed. Chen, Louis H. Y. et al., to appear.Google Scholar
[10] Hida, T., Lee, K.-S. and Lee, S.-S., Conformal invariance of white noise, Nagoya Math. J., 98 (1985), 87–98.Google Scholar
[11] Hida, T., Lee, K.-S. and Si, Si, Multidimensional parameter white noise and Gaussian random fields, Balakrishnan Volume (1987), 177–183.Google Scholar
[12] Hida, T., Si, Si, Variational calculus for Gaussian random fields, Proc. IFIP, Warsaw, 1988; Stochastic Systems and Optimization, Lecture Notes in Control and Information Sci. no. 136, ed. Zabczyk, J. (1989), 86–97.Google Scholar
[13] Hida, T., Kuo, H.-H., Pottoff, J. and Streit, L., White noise: an infinite-dimensional calculus, monograph in preparation.Google Scholar
[14] Kubo, I. and Takenaka, S., Calculus on Gaussian white noise, I-IV, Proc. Japan Academy A, Math. Sci., 56 (1980) 376–380, 411–416, 57 (1981) 433–437, 58 (1982) 186–189.Google Scholar
[15] Noda, A., Generalized Radon transform and Lévy’s Brownian motion I, II, Nagoya Math. J., 105 (1987), 71–87 and 89–107.Google Scholar
[16] Seeley, R. T., Spherical harmonics, Amer. Math. Monthly, 73 (1966), 115–121.Google Scholar
[17] Si, Si, A note on Lévy’s Brownian Motion, I, II, Nagoya Math. J., 108 (1987), 121–130 and 114 (1989), 165–172.Google Scholar
[18] Strichartz, R. S., Radon inversion-variations on a theme, Amer. Math. Monthly, 89 (1982), 377–384 and 420–423.Google Scholar
[19] Yosida, K., Lectures on differential and integral equations, Interscience Pub., 1960.Google Scholar